Problem 62 - Algebra and Trigonometry

Question

\(61-66\). Graph the three functions on a common screen. How are the graphs related? $$ y=x, \quad y=-x, \quad y=x \cos x $$

Step-by-Step Solution

Answer

The graphs of \(y = x\) and \(y = -x\) are straight lines reflected across the x-axis; \(y = x \cos x\) oscillates around them, increasing in amplitude away from the origin. Graphs cross at points where \(\cos x = \pm 1\).

Step 1: Identify the Functions to Graph

We have three functions to graph: \(y = x\), \(y = -x\), and \(y = x \cos x\). Each of these is a line or curve that we will analyze for their relations on a common coordinate plane.

Step 2: Graph the Function y = x

The function \(y = x\) is a straight line that passes through the origin \((0,0)\) with a slope of 1. This line rises one unit vertically for every one unit it moves horizontally, forming a 45-degree angle with the x-axis.

Step 3: Graph the Function y = -x

The function \(y = -x\) is a straight line that passes through the origin \((0,0)\) with a slope of -1. This line falls one unit vertically for every one unit it moves horizontally, forming a 45-degree angle with the x-axis but in the opposite direction to \(y = x\).

Step 4: Graph the Function y = x \cos x

The function \(y = x \cos x\) is not a straight line; it oscillates around the x-axis. It inherits the frequency of the \(\cos x\) curve but is scaled by \(x\), causing increasing amplitudes as \(x\) increases or decreases.

Step 5: Analyze the Relationships Between the Graphs

The graphs of \(y = x\) and \(y = -x\) are straight lines that are reflections of each other across the x-axis. The graph \(y = x \cos x\) oscillates about \(y = x\) and \(y = -x\), crossing these lines at the points where \(\cos x = 1\) or \(\cos x = -1\), respectively.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions you will encounter in mathematics. They are defined by the equation of the form \( y = ax + b \), where \( a \) and \( b \) are constants. In this general equation, \( a \) represents the slope, and \( b \) represents the y-intercept. This function graphs as a straight line, making it quite unique in its predictability.

Key characteristics of linear functions include:

• Constant slope: The rate of change between any two points on the line is the same.
• Straight line: The graph of a linear function is always a straight line.
• Symmetry in intercepts: If the y-intercept is zero, the line passes through the origin, otherwise it crosses the y-axis at the point (0, b).

In our exercise, \( y = x \) and \( y = -x \) are both examples of linear functions, with the lines perfectly representing the constant slope properties. These lines make angles of 45 degrees with the x-axis.

Slope

The slope is a central concept in understanding linear functions and is denoted by \( m \). It tells us how steep the line is and the direction in which it moves.

For the function \( y = x \), the slope \( m \) is 1. This means:

• For every unit increase along the x-axis, the y value increases by one unit.
• The line rises at a 45-degree angle because the x and y changes are equal.

On the other hand, for the function \( y = -x \), the slope \( m \) is -1:

• For every unit increase along the x-axis, the y value decreases by one unit.
• Reflects over the x-axis from \( y = x \) and shifts downward with the same tilt.

Grasping slope helps in predicting the behavior of a line on a graph just by knowing its equation.

Cosine Function

The cosine function is a periodic trigonometric function, often denoted as \( y = \cos x \). It regularly oscillates between -1 and 1, creating a wave-like appearance on graphs.

Some important features of the cosine function include:

• Periodicity: It has a period of \( 2\pi \), repeating its values every \( 2\pi \) units.
• Amplitude: The maximum height from its central axis is 1, meaning it oscillates between values +1 and -1.
• Symmetry: It is an even function, meaning \( \cos(-x) = \cos(x) \).

In the function \( y = x \cos x \), it is the cosine part that creates an oscillation. However, because it is multiplied by \( x \), the wave stretches larger as \( x \) increases.

Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states.

With our function \( y = x \cos x \), the graph does not stay constant in amplitude:

• Oscillation represents the wave-like fluctuations caused by the cosine component.
• The amplitude of these oscillations expands as \( x \) moves away from zero, due to its multiplication with \( x \).
• This means that the wave stretches out further as it moves towards positive or negative infinity, demonstrating both linear growth and periodic behavior.

Understanding oscillation is fundamental for analyzing behaviors that simulate waves, such as sound or light waves, and, in this case, the combination of linear growth with periodic behavior.